The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs

Yi-Zheng Fan, Tao Huang, Yan-Hong Bao, Chen-Lu Zhuan-Sun, Ya-Ping Li

Published 2017-04-28Version 1

Let $\mathcal{A}$ be a weakly irreducible nonnegative tensor with spectral radius $\rho(\mathcal{A})$. Let $\mathfrak{D}$ (respectively, $\mathfrak{D}^{(0)}$) be the set of normalized diagonal matrices arising from the eigenvectors of $\mathcal{A}$ corresponding to the eigenvalues with modulus $\rho(\mathcal{A})$ (respectively, the eigenvalue $\rho(\mathcal{A})$). Then $\mathfrak{D}$ is a finite abelian group containing $\mathfrak{D}^{(0)}$ as a subgroup, which acts transitively on the set $\{e^{\mathbf{i} \frac{2 \pi j}{\ell}}\mathcal{A}:j =0,1, \ldots,\ell-1\}$, where $|\mathfrak{D}/\mathfrak{D}^{(0)}|=\ell$ and $\mathfrak{D}^{(0)}$ is the stabilizer of $\mathcal{A}$. The spectral symmetry of $\mathcal{A}$ is characterized by the group $\mathfrak{D}/\mathfrak{D}^{(0)}$, and $\mathcal{A}$ is called spectral $\ell$-symmetric. By analyzing the property of $\mathfrak{D}$, we obtain some structural information of $\mathcal{A}$ by the spectral symmetry, especially for connected hypergraphs we get some results on the edge distribution and coloring. In particular, if $\mathcal{A}$ is also symmetric, we prove that $\mathcal{A}$ is spectral $\ell$-symmetric if and only if it is $(m,\ell)$-colorable. We characterize the spectral $\ell$-symmetry of a tensor by using its generalized traces, and show that for each positive integer $\ell$ with $\ell \mid m$, there always exists an $m$-uniform hypergraph $G$ such that $G$ is spectral $\ell$-symmetric.